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Global Financial Management

Forward and Future Contracts

Copyright 1997 by Campbell R. Harvey and Stephen Gray.
All rights reserved. No part of this lecture may be reproduced without
the permission of the authors.

Latest Revision: February 15, 1997

4.0 Overview:

This class provides an overview of forward and futures
contracts. Forwards and futures belong
to the class of securities known as derivatives since their value
is derived from the value of some other security. The price of a foreign
exchange forward contract, for example, depends on the price of the underlying
currency and the price of a pork belly futures contract depends on the
price of pork bellies. Derivatives trade both on exchanges (where contracts
are standardized) and over-
the-counter (where the contract specification can be customized). The focus
of this class is on

(1) definitions and contract specifications of the major exchange-traded
derivatives, (2) the mechanics of buying, selling, exercising, and settling forward
and futures contracts, (3) derivative trading strategies including hedging, and (4) the relationships between derivatives, the underlying security,
and riskless bonds.
In particular, it is possible to form combinations
of derivatives and the underlying
security that are riskless, providing a means
of valuing derivatives.

4.1 Objectives

After completing this class, you should be able to:

Determine the possible payoffs of forward and futures contracts.

Understand the mechanics of buying, selling, exercising, and settling
forward and futures contracts.

Determine the possible payoffs of portfolios of futures, forwards,
and the underlying asset.

Understand the directional effects of relevant variables on the value
of derivative securities.

Use standard valuation techniques to determine the price of forward
and futures contracts.

4.2 Introduction

Despite the recent adverse press they have received, derivative securities
provide a number of useful functions in the areas of risk management and
investments. In fact, derivatives were originally designed to enable market
participants to eliminate risk. A wheat farmer, for example, can fix a
price for his crop even before it is planted, eliminating price risk. An
exporter can fix a foreign exchange rate even before beginning to manufacture
the product, eliminating foreign exchange risk. If misused, however, derivative
securities are also capable of dramatically increasing risk.

This module focuses on the mechanics of forward and futures
contracts. There is a particular emphasis on the interrelationship between
the various contracts and the spot price of the underlying asset.
The spot price is the price of an asset where the sale transaction and
settlement is to occur immediately.

4.3 Forward Contracts

The Mechanics of Forward Contracts

A Forward Contract is a contract made today for delivery of an
asset at a prespecified time in the future at a price agreed upon today.
The buyer of a forward contract agrees to take delivery of an underlying
asset at a future time, T, at a price agreed upon today. No money
changes hands until time T. The seller agrees to deliver the underlying
asset at a future time, T, at a price agreed upon today. Again,
no money changes hands until time T.

A forward contract, therefore, simply amounts to setting a price today
for a trade that will occur in the future. Example 4.4 illustrates the
mechanics of a forward contract. Since forward contracts are traded over-the-counter
rather than on exchanges, the example illustrates a contract between a
user and a producer of the underlying commodity.

Example 4.4: Forward contract mechanics.

A wheat farmer has just planted a crop that is expected to yield 5000
bushels. To eliminate the risk of a decline in the price of wheat before
the harvest, the farmer can sell the 5000 bushels of wheat forward. A miller
may be willing to take the other side of the contract. The two parties
agree today on a forward price of 550 cents per bushel, for delivery five
months from now when the crop is harvested. No money changes hands now.
In five months, the farmer delivers the 5000 bushels to the miller in exchange
for $27,500. Note that this price is fixed and does not depend upon
the spot price of wheat at the time of delivery and payment.

4.5 Valuation of Forward Contracts

Forward contracts can be valued by recognizing that, in many cases,
forward markets are redundant. This occurs when the payoff from a forward
contract can be replicated by a position
in (1) the underlying asset and (2) riskless bonds. Before illustrating
this concept, we define the cost of carry of the underlying commodity. This is the cost involved
in holding a physical quantity
of the commodity. For wheat the cost of carry is a storage cost, for live
hogs it consists of storage and feed costs, and for gold it consists of
storage and security costs. Some commodities have a negative cost of carry.
For example, holding a stock index provides the benefit of receiving
dividends. In forward markets it is common to express the cost of carry
as a continuously compounded annual rate, payable at inception. For example,
if the cost of carry for wheat is reported to be 5%, this would mean that
the cost of storing $100 of wheat for six months is $100 (e0.05(0.5)-1) = $2.53, payable immediately.

We use the following notation, which is common in forward markets:

St

The spot price of the underlying commodity at
time t

S0

the spot price now, which is known.

ST

The spot price at maturity of the contract and
is not known when the contract is entered into.

r

The riskless rate of interest from now until maturity
of the contract,

q

The cost of carry of the underlying commodity

F

The forward price for delivery at time T.

Both r and q are expressed as continuously compounded
annual rates. Consider the strategy of:

borrowing enough money to buy one unit of a commodity and to pay for
the associated carrying costs through time T, and

entering into a forward contract to sell the commodity at time T.
The value of this position in terms of the initial (time 0) and
terminal (time T) cash flows is tablulated in the following table.

Arbitrage relationship between spot and forward contracts

Position

Initial Cash Flow

Terminal Cash Flow

Buy one unit of commodity

-S0

ST

Pay Cost of Carry

-S0(eqT - 1)

0

Borrow

S0 eqT

-S0 e(q+r)T

Enter 6-month forward sale

0

F - ST

Net Portfolio Value

0

F - S0 e(q+r)T

Since this portfolio requires no initial cash outlay, the absence of
arbitrage opportunities will ensure that the terminal payoff is also zero.
Therefore, the futures contract can be valued as

F = S0 e(q+r)T

The following example shows how arbitrage is possible if this pricing
relation is violated.

Example 4.6: Forward arbitrage.

Suppose the spot price of wheat is 550 cents per bushel, the six-month
forward price is 600, the riskless rate of interest is 5% p.a., and the
cost of carry is 6% p.a. To execute an arbitrage, you borrow money, buy
a bushel of wheat, pay to store it, and sell it forward. The cash flows
are:

That is, it is possible to lock in a sure profit that requires no intial
cash outlay.

4.7 Hedging With Forward Contracts

The primary motivation for the use of forward contracts is risk management.
The wheat farmer in example 4.4 was able to eliminate price risk by selling
his crop forward. Example 4.8 contains a more comprehensive example concerning
foreign exchange risk management.

Example 4.8: Forward contacts and risk management.

XYZ is a multinational corporation based in the US. Its manufacturing
facilities are located in Pittsburgh and hence its labor and manufacturing
costs are incurred in US dollars (USD). A large fraction of its sales,
however, are made to German customers who pay for the goods in Deutschemarks
(GDM).

There is a six-month lead time between the placement of a customer order
and delivery of the product. XYZ's cost of production is 80% of the sale
price. Suppose XYZ receives a $1MM GDM order and that the current USD/GDM
exchange rate is 0.60 (i.e. 1 GDM = 0.60 USD). The cost of production of
this order is $480,000 (0.60 x $1MM x 0.80). The exchange rate six months
from now is, of course, uncertain in which case XYZ is exposed to exchange
rate risk.

If the exchange rate stays at 0.60, then XYZ will convert the 1MM GDM
to $600,000 and earn a 25% profit on the $480,000 cost of production. If,
however, the exchange rate falls to 0.40 six months from now, XYZ will
convert the 1MM GDM to only $400,000, registering a loss on the sale.

Conversely, if the exchange rate rises to 0.80 six months from now,
XYZ will convert the 1MM GDM to $800,000, registering a very large profit
on the sale. Whereas XYZ are very good at manufacturing and marketing their
product, they have no expertise in forecasting exchange rate movements.
Therefore, they want to avoid the exchange rate risk inherent in this transaction
(i.e., the risk that they do everything right and then lose money on the
sale, solely because exchange rates move against them). They can do this
by selling forward 1MM GDM. This involves entering a contract today
with, say, an investment bank under which XYZ agrees to deliver 1MM GDM
six months from now in exchange for a fixed number of US dollars. This
rate of exchange is the six-month forward rate. Suppose the six-month
forward rate is 0.62 (which is set according to market expectations and
relative interest rates as described below). Then, when XYZ receives 1m
GDM from its customer, they deliver it to the investment bank in exchange
for $620,000 (locking in a profit) regardless of whether the exchange
rate happens to be 0.40 or 0.80 at that time.

4.9 Futures Contracts

The Mechanics of Futures Contracts

A futures contract is similar to a forward contract except for
two important differences. First, intermediate gains or losses are posted
each day during the life of the futures contract. This feature is known
as marking to market. The intermediate gains or losses are given
by the difference between today's futures price and yesterday's futures
price. Second, futures contracts are traded on organized exchanges with
standardized terms whereas forward contracts are traded over-the-counter
(customized one-off transactions between a buyer and a seller).

Example 4.10 illustrates the marking to market mechanics of the
All Ordinaries Share Price Index (SPI) futures contract on the Sydney Futures
Exchange. The SPI contract is similar to the Chicago Mercantile Exchange
(CME) S&P 500 contract and the London International Financial Futures
Exchange (LIFFE) FTSE 100 contract. The mechanics are the same for all
of these contracts. Stock index futures were introduced in Australia in
1983 in the form of Share Price Index (SPI) futures which are based on
the Australian Stock Exchange's (ASX) All Ordinaries Index which is the
benchmark indicator of the Australian stock market. Users of SPI futures
include major international and Australian banks, fund managers and other
large investment institutions. SFE locals and private investors are also
active participants in the market.

SPI futures have an underlying of A$25 x Index (ie., a SPI futures contract
with a price of 2000.00 will have a contract value of A$50,000). The All
Ordinaries Share Price Index (AOI) is a capitalisation weighted index and
is calculated using the market prices of approximately 318 of the largest
companies listed on the Australian Stock Exchange (ASX). The aggregate
market value of these companies totals over 95% of the value of the 1,186
domestic stocks listed.

Example 4.10: Marking to market.

Suppose an Australian futures speculator buys one SPI futures contract
on the Sydney Futures Exchange (SFE) at 11:00am on June 6. At that
time, the futures price is 2300. At the close of trading on June 6, the
futures price has fallen to 2290 (what causes futures prices to move is
discussed below).

Underlying one futures contract is $25 x Index, so the buyer's position
has changed by $25(2290-2300)=-$250. Since the buyer has bought the futures
contract and the price has gone down, he has lost money on the day and
his broker will immediately take $250 out of his account. This immediate
reflection of the gain or loss is known as marking to market.

Where does the $250 go? On the opposite side of the buyer's buy order,
there was a seller, who has made a gain of $250 (note that futures trading
is a zero-sum game - whatever one party loses, the counterparty gains).
The $250 is credited to the seller's account. Suppose that at the close
of trading the following day, the futures price is 2310. Since the buyer
has bought the futures and the price has gone up, he makes money. In particular,
$25(2310-2290)=+$500 is credited to his account. This money, of
course, comes from the seller's account.

This concept of marking to market is standard across all major
futures contracts. Contracts are marked to market at the close of trading
each day until the contract expires. At expiration, there are two different
mechanisms for settlement. Most financial futures (such as stock index,
foreign exchange, and interest rate futures) are cash settled, whereas
most physical futures (agricultural, metal, and energy futures) are settled
by delivery of the physical commodity. Example 4.11illustrates cash settlement.

Example 4.11: Cash settlement.

Suppose the SPI futures contact price was 2350 at the close of trading
on the day before expiration and 2360 at the close of trading on the expiration
day. Settlement simply involves a payment of $25(2360-2350) = $250 from
the seller's account to the buyer's account. The expiration day is treated
just like any other day in terms of standard marking to market.

An alternative to cash settlement is physical delivery. Consider the
SFE wool futures contract which requires delivery of 2500 kg of wool when
the contract matures. Of course, there are different grades of wool, so
a set of rules governing deliverable quality is required. These
are detailed rules that govern the standard quality of the underlying commodity
and a schedule of discounts and premiums for delivery of lower and higher
quality respectively. Example 4.12 illustrates the rules governing deliverable
quality for the SFE greasy wool futures contract.

Example 4.12: Deliverable quality:
Greasy wool futures.

Delivery must be made at approved warehouses in the major wool selling
centres throughout Australia. For wool to be deliverable, it must possess
the relevant measurement certificates issued by the Australian Wool Testing
Authority (AWTA) and appraisal certificates issued by the Australian Wool
Exchange Limited (AWEX). In particular, it must be good topmaking merino
fleece with average fibre diameter of 21.0 microns, with measured mean
staple strength of 35 n/ktx, mean staple length of 90mm, of good colour
with less than 1.0% vegetable matter. Because any particular bale of wool
is unlikely to exactly match these specifications, wool within some prespecified
tolerance is deliverable. In particular, 2,400 to 2,600 clean weight kilograms
of merino fleece wool, of good topmaking style or better, good colour,
with average micron between 19.6 and 22.5 micron, measured staple length
between 80mm and 100mm, measured staple strength greater than 30 n/ktx,
less than 2.0% vegetable matter is deliverable. Premiums and discounts
for delivery that does not match the exact specifications of the underlying
contract are fixed on the Friday prior to the last day of trading for all
deliverable wools above and below the standard, quoted in cents per kilogram
clean.

Example 4.13 illustrates the process of physical delivery for the SFE
greasy wool futures contract. The process is similar for most commodity
futures contracts.

Example 4.13: Physical delivery.

Suppose the greasy wool futures contact price was 700 cents at
the close of trading on the expiration day. Settlement involves physical
delivery, from the seller of the futures contract to the buyer, of the
underlying quantity of wool (2,500 kilograms) on the business day following
the expiration day. Delivery, therefore, involves the seller delivering
2,500 kg of wool to the buyer, in return for a payment of A$17,500.

The wool must be within the tolerance described above. If the wool is
of better quality than is specified in the contract, a premium must be
paid. Conversely wool of lower quality involves a discount. It is the seller
of the futures who must make delivery of the wool and he has the option
to choose what quality he will deliver, subject to the schedule of discounts
and premiums.

4.14 Margin

Although futures contracts require no initial investment, futures exchanges
require both the buyer and seller to post a security deposit known as margin.
Margin is typically set at an amount that is larger than ususal one-day
moves in the futures price. This is done to ensure that both parties will
have sufficient funds available to mark to market. Residual credit risk
exists only to the extent that (1) futures prices move so dramatically
that the amount required to mark to market is larger than the balance of
an individual's margin account, and (2) the individual defaults on payment
of the balance. In this case, the exchange bears the loss so that participants
in futures markets bear essentially zero credit risk. Margin rules are
stated in terms of initial margin (which must be posted when entering
the contract) and maintenance margin (which is the minimum acceptable
balance in the margin account). If the balance of the account falls below
the maintenance level, the exchange makes a margin call upon the
individual, who must then restore the account to the level of initial margin
before the start of trading the following day. Example 4.15 illustrates
the margining procedure.

Suppose a contract requires initial margin of $7,000 and maintenance
margin of $5,000. The following table illustrates the margining procedure
and the cash flows required for the buyer of a futures contract.

Time

Value of Futures Contract

Margin Balance before Calls

Margin Call

Margin Balance after Calls

0

25,000

0

7,000

7,000

1

24,000

6,000

0

6,000

2

22,000

4,000

3,000

7,000

3

24,500

7,000

0

7,000

Note that when the margin balance falls below the maintenance margin,
it must be restored to the initial level. Note also that when the futures
moves favorably (as at time 3) the marking to market cash inflow can be
immediately withdrawn - it need not remain in the margin account.

4.16 Valuation of Futures Contracts

Whereas the valuation of forward contracts is relatively straightforward,
the marking to market feature complicates the valuation of futures contracts.
The cash flows associated with forward and futures contracts are illustrated
in the following table.

Cash Flows of Forward and Futures Contracts

Time

0

1

2

...

T

Forward Cash Flow

0

0

0

...

ST-FO0

Futures Cash Flow

0

FU1-FU0

FU2-FU1

...

FUT - FUT-1

For both contracts, no money changes hands at the time the contract
is initiated (time 0). For the forward contract, no money changes
hand until the contract matures (time T). For the futures contract,
money changes hands daily depending upon movements in the futures price.

In some circumstances, however, a futures contract is perfectly equivalent
to a forward contract in which case the two contracts must have the same
value. Since forward contracts are relatively easy to value using a no-arbitrage
argument, this provides a convenient way of valuing a futures contract.
In particular, if interest rates are constant (at a continuously compounded
annual rate of r) over the life of the contract then the prices
of the futures contract and the forward contract are identical.

This equivalence can be established by considering a roll-over strategy
whereby at time 0 an investor purchases er futures
contracts and invests FU0 in a riskless bond where FU
represents the futures price). At time 1 the profit (possibly negative)
on the futures position is er(FU1-FU0),
which he invests (or borrows) until maturity. At maturity, this has grown
to er(T-1)er[FU1-FU0)] =
erT(FU1-FU0).

At time 1 he increases his holding to e2r contracts.
At time 2 the profit (possibly negative) on this position is e2r(FU2-FU1),
which he invests (or borrows) until maturity. At maturity, this has grown
to er(T-2)e2r(FU2-FU1) =
erT(FU2-FU1). At time 2 he
increases his holding to e3r and so on. At maturity,
the total payoff on the futures position is:

where we note that FUT = ST. The payoff
on the bond is FU0 e rT. Therefore, the overall
initial investment required for this strategy is $FU0
and the overall payoff at time T is S0erT.

Now consider the strategy of buying erT forward contracts
on day o and investing $FO0 in a riskless bond
(where FO represents the price of a forward contract). The overall
initial investment required for this strategy is $FO0
and the overall payoff at time T is:

erT(ST - FO0) + FO0
erT = S0 erT

Since both of these strategies have the same payoff, they must cost
the same. That is FO0 = FU0. The following
table illustrates the cash flows associated with the two strategies.

For the remainder of this module, we assume that interest rates are
indeed constant over the period of the contract and hence the futures price
equals the forward price. That is, we can consider the price and payoffs
of a futures contract to be identical to those of a forward contract. This
simplifies things because a forward contract has only a single payoff at
maturity.

Consider, for example, the valuation of a futures contract on the S&P
500 stock index. This contract, which trades on the Chicago Mercantile
Exchange (CME) entitles the buyer to receive the cash value of the S&P
500 stock index at the end of the contract period. There are always
four contracts in effect at any one time expiring in March, June, September,
and December. In contrast to the previous examples that involved a cost
of carry, holding the S&P 500 index yields a benefit, in the
form of dividends received, rather than a cost of carry. The result is
that the value of an S&P 500 futures contract can be expressed
as

F = S0 e (r-d)T

where

F

The Futures price

S0

The current value of the S&P 500 stock
index

r

The interest rate (annual continuously compounded
T-Bill rate)

d

The dividend yield on the index (continuously
compounded annual rate)

T

The time to maturity of the contract

This is the same as equation (1) except that "+q" has
been replaced by "-d" as the cost of carry (storing
wheat) has been replaced by a benefit (dividends). To see why this
relationship must hold, consider the strategy of (1) borrowing e-dTS0
through time T, (2) using this to purchase e-dT
units of the index and reinvesting all dividends back into the index, and
(3) selling a futures contract that matures at time T. If interest
rates are constant, the futures contract is equivalent to a forward contract,
which simplifies the analysis. In particular, the (equivalent) cash flows
associated with this strategy are tabulated in the following table. Note
that reinvestment of the dividends has resulted in the initial investment
of e-dT units of the index growing at a rate of d
to amount to one unit by maturity.

Examples 4.18 and 4.19 illustrate how to execute a riskless arbitrage
if this equality does not hold.

Arbitrage Relationship Between Spot and Futures Contract

Position

Time 0

Time T

Borrow

e-dTS0

erTe-dTS0

Buy e-dT units of index

-e-dTS0

ST

Sell one Futures Contract

0

F - ST

Net Position

0

F - S0e(r-dT)

Once again, since this strategy requires no initial cash outlay, the
cash flow at maturity must also be zero or an arbitrage opportunity exists.
In particular, if F > S0 e(r-dT) the strategy
of buying the index and selling the futures generates an arbitrage profit.
Conversely, if F < S0 e(r-dT) the strategy
of selling the index and buying the futures generates an arbitrage profit.

Examples 4.18 and 4.19 illustrate how to execute a riskless arbitrage
if this equality does not hold.

Example 4.18: Futures arbitrage: Buy index - Sell
futures

Suppose the S&P 500 stock index is at $295 and the
six-month futures contract on that index is at $300. If the prevailing
T-Bill rate is 7% and the dividend rate is 5%, an arbitrage
opportunity exists because F=300 > S e(r-d)T = 297.96.
The arbitrage can be executed by buying low and selling high. In this case,
the futures contract is relatively overvalued, so we sell the futures and
buy the index.

In particular, the strategy is to

Borrow e-dTS0 = $287.72 at 7% repayable
in 6 months.

Use this $287.72 to buy e-dT = 0.975 units
of the S&P index, and reinvest all dividends in the index.

Sell a futures contract for delivery of the index in six months.

This generates the following cash flows:

Position

Time 0

Time T

Borrow

287.72

-297.96

Buy e-dTunits of index

-287.72

ST

Sell one futures contract

0

300 - ST

Net Position

0

2.04

Hence this strategy generates an arbitrage profit of $2.04 six
months from now.

Example 4.19: Futures arbitrage: Sell index - Buy futures

Suppose the S&P 500 stock index is at $300 and the
six-month futures contract on that index is at $300. If the prevailing
T-Bill rate is 7% and the dividend rate is 5%, an arbitrage
opportunity exists because F = 300 < S e(r-d)T= 303.02. The arbitrage can be executed by buying low and selling
high. In this case, the futures contract is relatively undervalued, so
we buy the futures and sell the index.

In particular, the strategy is to

Short sell e-dT units of the S&P index
generating Se-dT = 292.59.

Lend the $292.59 proceeds of the short sale at 7% repayable
in 6 months.

Buy a futures contract for delivery of the index in six months.

This generates the following cash flows:

Position

Time 0

Time T

Sell e-dT units of index

292.59

-ST

Lend

-292.59

303.02

Buy one futures contract

0

ST-300

Net Position

0

3.02

Hence this strategy generates an arbitrage profit of $3.02 six
months from now.

4.20 Hedging with Futures

In this section, we examine how three common business risks - interest
rate risk, stock market risk, and foreign exchange risk - can be hedged
in a practical setting. In each case, we describe the nature of the risk
and illustrate, through a series of practical examples, how the risk can
be managed.

4.21 Hedging Interest Rate Risk

There are two primary interest rate futures contracts that trade on
US exchanges. The Eurodollar Futures Contract trades on the Chicago
Mercantile Exchange and the US T-Bill Futures Contract trades on
the Chicago Board of Trade.

The Eurodollar contract is the more successful and heavily traded contract.
At any point in time, the notional loan amount underlying outstanding Eurodollar
futures contracts is in excess of $4 trillion. This contract is based on
LIBOR (London Interbank Offer Rate), which is an interest rate payable
on Eurodollar Time Deposits. This rate is the benchmark for many
US borrowers and lenders. For example, a corporate borrower may be quoted
a rate of LIBOR+200 basis points on a short-term loan. Eurodollar time
deposits are non-negotiable, fixed rate US dollar deposits in banks that
are not subject to US banking regulations. These banks may be located in
Europe, the Carribean, Asia, or South America. US banks can take deposits
on an unregulated basis through their international banking facilities.
LIBOR is the rate at which major money center banks are willing to place
Eurodollar time deposits at other major money center banks. Corporations
usually borrow at a spread above LIBOR since a corporation's credit risk
is greater than that of a major money center bank. By convention, LIBOR
is quoted as an annualized rate based on an actual/360-day year (i.e.,
interest is paid for each day at the annual rate/360).

Example 4.22 demonstrates how interest is calculated on a LIBOR loan
according to the conventions.

Example 4.22: LIBOR Conventions.

If 3-month (90 actual days) LIBOR is quoted as 8%, the interest payable
on a $1 million loan at the end of the 3-month borrowing period is

(.08)(90/360) $1,000,000 = ((.08)/4) $1,000,000 = $20,000

The Eurodollar futures contract is based on a 3-month $1 million Eurodollar
time deposit. It is cash settled, so no actual delivery of the time deposit
occurs when the contract expires. Delivery months are March, June, September,
and December. The minimum price move is $25 per contract which is equivalent
to 1 basis point: (.0001/4)1,000,000=25. The futures price at expiration
(time T) is determined as FT = 100-LIBOR. Prior
to expiration, the futures price implies the interest rate that can be
effectively locked in for a 3-month loan that begins on the day the contract
matures. Settlement of the Eurodollar futures contract is illustrated in
Example 4.23.

Example 4.23: Settlement of Eurodollar
Futures Contract.

Suppose you purchased 1 December Eurodollar futures contract on November
15 when the price was 94.86. If interest rates fall 100 basis points between
November 15 and expiration of the futures contract in December, what is
your total gain or loss on the contract at settlement?

First note that no money changes hands at the time you buy the contract.
This is the nature of all futures contracts. The November 15 price of 94.86
implies that the LIBOR rate of interest was 100-94.86=5.14% at that time.
If LIBOR falls 100 basis points by the time the December contract expires,
LIBOR will then be 4.14%. Therefore, the expiration futures price will
be 100-4.14=95.86. The total gain is therefore:

0.25 (1,000,000)(Ft-F0)
= 0.25 (1,000,000)(0.9586-0.9486) = $2,500

That is, to settle the contract, your counterparty will give you $2,500.

Example 4.24 contains a detailed illustration of how the Eurodollar
futures contract can be used to hedge interest rate risk.

Example 4.24: Hedging with the
Eurodollar Futures Contract.

It is currently November 15 and your company is aware that it needs
to borrow $1 million on December 16 to pay a liability which falls due
on that day. The loan can be repaid on March 16 when an account receivable
will be collected. The current LIBOR rate is 5.14%. Your company is concerned
that interest rates will rise between now and December 16, in which case
you will pay a higher rate of interest on your loan. How can your company
lock in the current rate of 5.14%?

Your company stands to lose if interest rates increase. Therefore, you
want enter a futures position that increases in value if interest rates
rise. Then, if interest rates rise, your company loses by paying higher
interest charges on the loan, but your company gains by profiting on the
futures position. Conversely, if interest rates fall, your company gains
by paying lower interest charges on the loan, but your company loses on
the futures position. Ideally, the loss and the gain would exactly cancel,
whether interest rates rise or fall.

From the construction of the Eurodollar futures contract, we know that
if the interest rate rises, the futures price will fall. Therefore, you
will sell 1 December Eurodollar futures contract at 94.86. Underlying this
contract is a notional 3-month $1 million dollar loan to be entered into
on December 16 (the day the contract expires).

If we could lock in the rate of 5.14%, the total interest on the loan
would be 0.0514($1 million)/4 = $12,850.

First, suppose that on December, 16 LIBOR is 6.14%. Interest on the
loan will be
0.0614($1 million)/4 = $15,350, and the gain on the futures position will
be -10000(93.86-94.86)/4 = $2,500. This yields a net cash outflow of -$15,350+$2,500
= -$12,850, which is the same as 3-month's interest on $1 million at 5.14%.

Now suppose that on December, 16 LIBOR is 4.14%. Interest on the loan
will be
0.0414($1 million)/4 = $10,350, and the gain on the futures position will
be -10000(95.86-94.86)/4 = -$2,500. This yields a net cash outflow of -$10,350-$2,500
= -$12,850, which is the same as 3-month's interest on $1 million at 5.14%.

4.25 Hedging Market Risk

Another source of risk that an individual or organization may wish to
hedge is stock market risk. For example, a person nearing retirement may
wish to hedge the value of the equities component of his retirement fund
against a stock market crash before he retires. A fund manager, who believes
he can pick winners among individual stocks may wish to hedge market-wide
movements. The dominant stock market index futures contract is the S&P
500 futures contract. This contract trades on the Chicago Mercantile
Exchange and has delivery months March, June, September, and December.
The underlying quantity is $500 times the level of the S&P 500 index.
The minimum price move is 0.05 index points, which is $25 per contract.
Example 4.26 illustrates the settlement mechanics for the S&P 500 contract.

Example 4.26: Settlement of the
S&P 500 Futures Contract.

It is currently November 15 and the S&P 500 index is at 382.62.
The December S&P 500 futures price is 383.50. If you buy 1 December
S&P 500 futures contract, how much will you gain if the futures price
at expiration is $393.50?

The gain on your futures position is $500(Ft-F0)
= $500(393.50-383.50)=$5,000. That is, to settle the contract, your
counterparty will give you $5,000.

Example 4.27 contains a detailed illustration of how the S&P 500
futures contract can be used to hedge stock market risk.

Example 4.27: Hedging with the
S&P 500 Futures Contract.

A portfolio manager holds a portfolio that mimics the S&P 500 index.
The S&P 500 index started the year at 306.8 and is currently at 382.62.
The December S&P 500 futures price is currently 383.50. The manager's
fund was valued at $76.7 million at the beginning of the year. Since the
fund has already generated a handsome return for the year, the manager
wishes to lock in its current value. That is, he is willing to give up
potential increases in order to ensure that the value of the fund does
not decrease. How does he lock in the current value of the fund?

First note that at the December futures price of 383.50, the return
on the index, since the beginning of the year, is 383.5/306.80-1 = 25%.
If the manager is able to lock in this return on his fund, the value of
the fund will be 1.25($76.7 million) = $95.875 million. Since the
notional amount underlying an S&P 500 futures contract is 500(383.50)
= $191,750, the manager can lock in the 25% return by selling 95,875,000/191,750=500
contracts. To illustrate that this position does indeed form a perfect
hedge, we examine the net value of the hedged position under two scenarios.

First, suppose the value of the S&P 500 index is 303.50 at the end
of December. In this case, the value of the fund will be (303.50/383.50)95.875
million = 75.875 million. The gain on the futures position will be
-500(500)(303.50-383.50) = 20 million. Hence the total value of
the hedged position is 75.875+ 20 = 95.875 million, locking in a 25% return
for the year.

Now suppose that the value of the S&P 500 index is 403.50 at the
end of December. In this case, the value of the fund will be 403.50/383.50(95.875
million) = 100.875 million. The gain on the futures position will be
-$500(500)(403.50-383.50) = -5 million. Hence the total value of
the hedged position is 100.875-5=95.875 million, again locking in a 25%
return for the year.

4.28 Hedging Foreign Exchange Risk

Another source of risk that an individual or organization may wish to
hedge is foreign exchange risk. For example, a person who will be travelling
overseas in the coming months may wish to hedge the value of the amount
of money he intends to spend abroad against a devaluation of his domestic
currency relative to the foreign currency. An exporter who sells goods
overseas on credit may wish to hedge against a devaluation of the foreign
currency in which payment occurs.

A number of foreign currency futures contracts trade on the International
Monetary Market division of the Chicago Mercantile Exchange. The currencies
on which contracts are based, and the underlying notional amount are listed
in the following Table. Delivery months for all contracts are March, June,
September, and December. Prices are quoted as US dollars per unit of foreign
currency. For example, if one Swiss franc buys 69.15 US cents, the price
will be quoted as 0.6915.

Your company sells 10 machines to a Swiss company. The sale price is
100,000 Swiss Francs each and payment is to be made at the end of the calendar
year. The December futures price for Swiss Francs is 0.6915. You are worried
that the Swiss Franc will depreciate against the US Dollar between now
and the end of the year. How can you hedge this exchange rate risk?

Note that since (1) the total exposure is one million Swiss Francs and
(2) each futures contract is for 125, 000 Francs, eight contracts are required
to hedge the exposure. Further, since (1) the company stands to lose if
the Swiss Franc depreciates (each Swiss Franc can be converted back into
a smaller number of Dollars) and (2) the futures contracts decrease in
value if the Swiss Franc depreciates (since the basis of the contract is
Swiss Francs per Dollar), the contracts should be sold.

To illustrate that selling eight futures contracts provides an adequate
hedge, first suppose that the value of the Swiss Franc is 0.30 at the end
of December. In this case, the US Dollar value of the payment for the machines
will be 0.30(10)(100,000) = $300,000. The gain on the futures position
will be -8(125,000)(0.30-0.6915) = $391,500. Hence the total income is
$691,500, which equals the unhedged income in dollars if the exchange rate
does not fluctuate.

4.30 Basis Risk

There is no such thing as a perfect hedge. You can never completely
eliminate a cash position's risk. Consider a holder of Q Treasury
bonds maturing in 2004 with a coupon rate of 8%. Assume that the holder
of bonds believes that bond prices are going to fall. To hedge his risk,
the person shorts an equivalent amount of futures contracts for Treasury
bonds. At a later date, the person will close out both its bond and futures
positions. At the close, the firm will receive BT per
bond sold in the regular spot or cash market. The futures
price is F0 at the time the futures are sold short, and
its price at the closeout is FT. Prior to the closeout,
both BT and FT are uncertain, although
F0 is known. The usual computation of the funds that
the person will have at closeout is:

From the above equation, the net revenue from the hedge position is
composed of (1) a certain component that depends upon the futures price
at the time of the hedge (F0) and (2) an uncertain component
that depends upon the difference between the price received for bonds in
the spot market and the futures price at closeout (BT-FT).
The difference between the spot and the futures price is called the basis.
Thus, uncertainty about the net hedged revenue arises if there is uncertainty
about the basis. To quote Holbrook Working, "hedging is speculation
in the basis".

There are many reasons for the basis to be uncertain.

First, the good or instrument being hedged may be different from the
good or instrument for which there is a futures contract. This would be
the case if a corporate bond offering is hedged with Treasury bond futures;
basis risk arises due to the uncertainty of the yield differential at the
time the hedge is lifted.

Second, in commodity futures, there is basis risk due to locational
differentials. For example, a cattle farmer in Texas who hedges with a
cattle futures contract that calls for delivery in Omaha has the uncertainty
of the closeout differential between the Texas steer price and the Omaha
steer price. This is called locational basis risk. This is usually
an important factor in agricultural contracts. The risk is compounded by
the fact that the seller usually has the option of where delivery is made.

The third type of basis risk arises because the seller of the futures
contract often has the option to choose the quality of the goods
or financial instrument delivered. For example, the Treasury bond futures
market calls for delivery of any U.S. Treasury bond that is not callable
within 15 years. Since there are many instruments that are candidates for
delivery, the hedge has the risk of fluctuations in the yield spread between
the instrument hedged and the instrument ultimately delivered.

Fourthly, with most futures contracts, the seller has the choice of
the date of delivery within the delivery month. This choice is an uncertain
value and thus contributes to basis risk.

Finally, the mark to market aspect of futures results in hedging risk.
The uncertainty is about the amount of interest earned or forfeited due
to the daily transfers of profits and losses. In fact, the equations for
net revenue are not exactly right due to the omission of interest earned
(lost) on futures profits (losses).

4.31 The Volatility of Futures

A common mistake made is to assume that futures are much more volatile
than stocks. Percentage changes of futures prices are generally less volatile
than the percentage changes of a typical stock. Annualized standard deviations
for most futures contracts are in the 15-20% range whereas a typical stock's
is about 30%.

There is no reason that the futures should be played in a high risk
manner by a large investor. Of course, if the futures investor does not
have enough capital (5-8 times margin), then he is required to play with
considerable leverage or not at all. Before taking great leverage, the
small investor should consider looking at a smaller contract (grain on
CBT is 5,000 bushels whereas Mid-America contract is 1,000 bushels).

The effect of leverage is to increase volatility. Borrowing to meet
the margin requirements will increase gains but also increase losses. Setting
aside larger amounts of capital which are invested in a safe asset will
decrease the volatility.

4.33 Risk in the Futures Markets

As we have already seem, one the most important applications of the
futures is for hedging. Futures contracts were initially introduced to
help farmers that did not want to bear the risk of price fluctuations.
The farmer could short hedge in March (agree to sell his crop) for
a September delivery. This effectively locks in the price that the farmer
receives. On the other side, a cereal company may want to guarantee in
March the price that it will pay for grain in September. The cereal company
will enter into a long hedge.

There are a number of important insights that should be reviewed. The
first is that we should be careful about what we consider the investment
in a futures contract. It is unlikely that the margin is the investment
for most traders. It is rare that somebody plays the futures with a total
equity equal to the margin. It is more common to invest some of your capital
in a money market fund and draw money out of that account as you need it
for margin and add to that account as you gain on the futures contract.
It is also uncommon to put the full value of the underlying contract in
the money market fund. It is more likely that the futures investor will
put a portion of the value of the futures contract into a money market
fund. The ratio of the value of the underlying contract to the equity invested
in the money market fund is known as the leverage. The leverage is a key
determinant of both the return on investment and on the volatility of the
investment. The higher the leverage -- the more volatile are the returns
on your portfolio of money market funds and futures. The most extreme leverage
is to include no money in the money market fund -- only commit your margin.

The second important insight had to do with hedging with futures contracts.
The concept of basis risk was introduced. It is extremely unlikely
that you can create a perfect hedge. A perfect hedge is when the loss on
your cash position is exactly offset by the gain in the futures position.
We suggested some reasons why it is unlikely that we can construct a perfect
hedge.

The most obvious case is when you are trying to hedge a cash position
with futures positions in different instruments. This is the case that
we introduced in one of the first lectures when we hold the Ginnie Mae
security and want to hedge this security with a combination of T-Bonds
and Euros. It is unlikely, however, that at the expiration of the futures
contract, the cash price of the T-Bond and Euros will equal the Ginnie
Mae. This is the basis risk.

A second type of basis risk arises out of the quality option.
We discussed this in terms of food and financial instruments. If you are
a farmer and want to lock in the price for your crop of wheat, you may
use a futures contract that may call for delivery of a number of different
types of wheat. Similarly, in the T-Bond and T-Note contracts, there are
a whole range of instruments that are available for delivery. This difference
will induce basis risk.

Third, there is a timing option. The futures contract is different
from an options contract. Most futures call for delivery within the contract
month. It is unclear when the short will deliver the goods. This uncertainty
leads to basis risk.

The fourth type of risk is locational basis risk. This is mainly applicable
to agricultural commodities. There could be a difference the cash price
of the good that you are selling (cattle) and the futures price at a different
location.

The last type of uncertainty is linked to the uncertain interest rate
flows from the money you make in excess of the margin.

4.34 More on Hedging

Since hedging is such an important application of futures contracts,
we have provided more
examples of hedging Some textbook examples which do the hedging incorrectly
are included, to show you some of the common pitfalls involved with hedging.

Summary of Important Formulas

F = S0 e(r+q)T

The price of a forward contract when there is a cost of carry q.
When interest rates are constant, the same relationship holds for a futures
contract.

F = S0 e(r-d)T

The price of a forward contract when there is a dividend benefit d.
When interest rates are constant, the same relationship holds for a futures
contract.